Convergence of Entropic Schemes for Optimal Transport and Gradient Flows

TL;DR

This paper proves the convergence of entropic regularization schemes for optimal transport problems and gradient flows, providing theoretical foundations for their use in computational and applied mathematics.

Contribution

It establishes the Γ-convergence of entropic regularized optimal transport to the Monge-Kantorovich problem and shows convergence of entropic gradient flows to the original flows as regularization vanishes.

Findings

Γ-convergence of entropic optimal transport to Monge-Kantorovich

Convergence of entropic gradient flows to original flows

Theoretical validation of entropic schemes in optimal transport

Abstract

Replacing positivity constraints by an entropy barrier is popular to approximate solutions of linear programs. In the special case of the optimal transport problem, this technique dates back to the early work of Schr\"odinger. This approach has recently been used successfully to solve optimal transport related problems in several applied fields such as imaging sciences, machine learning and social sciences. The main reason for this success is that, in contrast to linear programming solvers, the resulting algorithms are highly parallelizable and take advantage of the geometry of the computational grid (e.g. an image or a triangulated mesh). The first contribution of this article is the proof of the $\Gamma$-convergence of the entropic regularized optimal transport problem towards the Monge-Kantorovich problem for the squared Euclidean norm cost function. This implies in particular the convergence of the optimal entropic regularized transport plan towards an optimal transport plan as the entropy vanishes. Optimal transport distances are also useful to define gradient flows as a limit of implicit Euler steps according to the transportation distance. Our second contribution is a proof that implicit steps according to the entropic regularized distance converge towards the original gradient flow when both the step size and the entropic penalty vanish (in some controlled way).